Vol-7, Special Issue3-April, 2016, pp372-384
http://www.bipublication.com
Research Article
A new method for solving fredholm integral equation
Mohammdali Zafarmandi Ardabili, Abbas Zakaeieh*
and Khodadad Oraki
Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran
*Corresponding author: Abbas Zakaeieh
ABSTRACT:
In this paper, we use hat basis function to solve. The system of Fredholm integral Equations (SFIEs) of the second kind. This method converts the system of integral equations into linear system of algebraic equations. Applications of the method on some examples show its accuracy and efficiency.
Key words:Fredholm integral equations, Hat functions, Vector forms, Triangular Function.
INTRODUCTION:
In recent years, many different methods and different basis functions have been used to estimate the solution of the system of integral equations, such as Adomian decomposition method, Taylor’s expansion method, projection method and Nystrom method, Spline collocatin method, Runge-Kutta method, Sinc method, Tau method, Block-Pulse functions and operational matrices. In the present paper, we use hat basis functions to solve the system of linear Fredholm integral equations of the second’s kinds:
(1-1)
,
,
Where and are in , belongs to . Moreover
and are known and is unknown. We assume that Eqs. (1-1) have unique solution.
Review of Block Pulse Functions and the Triangular Functions:
As proposed by many authors, specially in [10], a set of block-pulse functions (BPFs) are usually definded in the interval [0, 1) as
(2-1)
Where and is an arbitrary positive integer.According to (2-1), the unit
interval is divided into equidistant subintervals, and the Th Block-Pulse function has only
one rectangular pulse of unit heigh in the subinterval . In fact, the restriction of an interval with
(2-1). We can obtain a similardefinition of Block-Pulse functions with regard to an interval of arbitrary lengh to meet the requirements of various problems. Therefore for we can define the
Block-Pulse functions in the interval as
Seed planting were taken in each pot at a depth of 2-3 can above the soil surface. Organic and chemical fertilizers consumed. Where is an aribitrary positive integer?
In the this paper, for convenience, we consider the Block-Pulse functions in the interval , and denote
the width of Block-Pulse functions as . One of the important properties of Block-Pulse functions is
the disjointness of them. Which can directly be obtained from
(2-3)
Where . The orthogonality of Block-Pulse functions is derived immediately form
(2-3)
Where is the Kronecker delta given by for and for .
The set of BPFs is complete when approaches infinity, i.e, for every . Perceval’s identity holds as follows:
Where , Where
,
,
,
And
,
(2-4)
In vector form, an arbitrary function can be expanded in the form ,
Where with , defined in (2-4) and
.
Where ,
Now, let be a function of two variables defined on . Then can be
expanded by BPFs , where and are and dimensional
Block Pulse coefficient matrix.
, for and
. In this paper we always consider , even in the next section,
for expanding by TFs. Form the distinct property of BPFs. The multiplication of two BPF vectors is as follows:
. (2-5)
The operational matrix for integration of BPFs has been derived as the following upper triangular matrix:
,
, (2-6)
The matrix P performs as an integrator, and it is pivotal in any BPF analysis, thus
, (2-7)
Now, we demonstrate the construction of triangular functions (TFs) according to
(2-8)
Where is defined as
(2-9)
And is defined as
(2-10)
Therefore, for each , and can be constructed as
above. Hence, we can generate two vectors of orthogonal TFs. namely such that
It could be said that these two vectors are complementary to each other as far as BPFs are considered. The authors of [1] called the left-handed triangular functions and the right-handed triangular
functions. Now, if we divide the interval to equal parts, we have
, (2-12)
(2-13)
We can approximate the function by as follows
+… +
(2-14)
Note 2.1[1]. It is apparent form (2-4) and (2-14) that unlike BPFs the representation by TFs dose not need any integration to evaluate the cofficients, thereby reducing a lot of computational efforts.
The orthogonality of LHTF set (similarly RHTF set) resulted from mutual disjointness of LHTF (and
RHTF), i.e. for .
and (2-15)
The operational matrix for integaration has been computed in Deb et al.[1]. They proved that
, (2-16)
Where
and (2-17)
One can similarly show that the integration operational matrix of is the same as (2-16) and (2-17). If we denote the integration operational matrix of BPF by P as in (2-6) then it can be easily seen that . For more details about triangular orthogonal functions, see [1].
3.Some new properties of TFs
The following properties of the product of two vectors will be used:
, (3-1)
, (3-2)
(3-4)
where is the zero matrix. First notice that, functions and also can be represented by
, (3-5)
, (3-6) for
, where is the unit step function defined as
(3-7)
Now, we show that (3-2) holds Eq.(3-1) can be obtained similarly. For it, consider the product , it is sufficient to show that for each
.
By disjointness of and for , the desired result will be obtained
=
(3-8)
Notice that, we can approximate the result by and in (3-8),
is possibly nonzero only in . Thus, the last relation in
(3-8) is followed. We now show that Eq. (3-3)holds. For it
Where , thus
,
Similar argument discussed for (3-8) results that
(3-9)
, Other properties that we will need are
(3-11)
Where is identity matrix. To show (3-10), consider (3-8) again
. (3-12)
Similarly, , and one can easily prove that (3-11) holds.
Note 3.1. the obtained properties in this section very familiar for us and also they are compatible to BPF properties. Form (2-17) and (2-6), we have
(3-13) And from (2-5),(2-9) and (3-1),(3-2)
,
,
= ,
,
, (3-14)
Note 3.2. Where diag( for each vector is a diagonal matrix that its entries are the componets of
vector . Furhemore, since , for each form
(3-10) and (3-13), we have
.
The equations mentioned above confirm the close relation between BPFs and TFs.
Theorem3.2. suppose that where and
are m-tuple vectors of coefficients then we have , . 4. Expanding two variable functions by TFs
We can expand each by two TFs vectors, with and
components. Respectively for convenience consider . For obtaining desired results, we
(4-1)
Now, each for for can be expanded by TFs with respect to independent
variables. Hence, the expansion of can be written
,
,
Which in equation above
(4-2)
(4-3)
(4-4)
(4-5)
5. Fredholm integral equation of the second kind
, (5-1)
Where and is the unknown
Function. Our problem is to determine TF Patroefficients of in the interval
from the know functions , and kernel . Let us expand and
by as follows:
(5-2)
(5-3)
as described in section 4, we can expand in the interval by TFs. suppose that this approximation be as follows:
(5-4).
where , are obtained from Eqs.(4-2)-(4-5). Then we have
thus, we have
by using define integral formula for TFs in ( 3-10) and (3-11). We have
(5-5)
By applying theorem 3.2, the coefficients of and on both sides of (5-5) must be equal, hence, we have the following equations for the coeffcients of TFs.
.
Set
, (5-6)
(5-7)
(5-8)
(5-9).
Thus, we have the following linear system:
.
After solving the above linear system. We can find and and then
is the estimation of the solution of the Fredholm integral equation of the second kind.
6. Hat basis functions (HFs)
A set of hat functions (HFs) are usually defined on as
(6-1)
For (6-2)
(6-3)
Where and is an arbitrary positive integer. Indeed the unit interval is divided into
equidistant subintervals. According to the definition of HFs
(6-4)
(6-6)
An arbitrary function can be expanded in vector form as:
(6-7)
Where and and
(6-8)
Now let be an arbitraryfunction of two variables defined on
. It can be expanded by HFs
(6-9)
Where and are and dimensional HFsvectors, respectively, and is
the . HFs coefficients matrix with entries
.
In this paper, for convenience, we put . Moreover relation (6-5) follows
(6-10)
(6-11)
Where is the following matrix,
(6-12)
7. A method to solve system of linear fredholm integral equation of the second kind: Consider (1.1). We consider equation of (1.1):
(7-1)
We approximate functions and with respect to HFs as follows:
(7-2)
Where vectors and and matrices are HFs coeffcients of , and ,respectively.
Substituting (7-2) into (7-1), we will have
(7-3) The system of linear equation (7-3),can be expressed in a matrix form:
, (7-5)
, (7-6)
Where and are dimensional vectors,and
. (7-7)
By solving the system of Eq.(7-4) we obtain unknowns and then
, (7-8) At this stage, the expression of convergence and order it to pay
Let be Banach space with norm
.
We define the integral operator as follows:
, Where
Therefore, we can write the system of integral equation (1.1) in the operator form .
Now , let interval , by choosing integer , be divided
into equal subintervals. We choose the subspace , the set of all functions that are
piecewise linear on , with breakpoints and dimension .indeed, all hat basis functions explained in
Stated before to Let . We introduce a continuous linear
projection operator such that
Where
We have [see more 1].
8.numerical example
and , and we have:
, ,
- , , and
.
The numerical results are shown in Table
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